2Vectors
IA Vectors and Matrices
2.3 Cauchy-Schwarz inequality
Theorem (Cauchy-Schwarz inequality). For all x, y ∈ R
n
,
|x · y| ≤ |x||y|.
Proof. Consider the expression |x −λy|
2
. We must have
|x − λy|
2
≥ 0
(x − λy) · (x − λy) ≥ 0
λ
2
|y|
2
− λ(2x · y) + |x|
2
≥ 0.
Viewing this as a quadratic in
λ
, we see that the quadratic is non-negative and
thus cannot have 2 real roots. Thus the discriminant ∆ ≤ 0. So
4(x · y)
2
≤ 4|y|
2
|x|
2
(x · y)
2
≤ |x|
2
|y|
2
|x · y| ≤ |x||y|.
Note that we proved this using the axioms of the scalar product. So this
result holds for all possible scalar products on any (real) vector space.
Example.
Let
x
= (
α, β, γ
) and
y
= (1
,
1
,
1). Then by the Cauchy-Schwarz
inequality, we have
α + β + γ ≤
√
3
p
α
2
+ β
2
+ γ
2
α
2
+ β
2
+ γ
2
≥ αβ + βγ + γα,
with equality if α = β = γ.
Corollary (Triangle inequality).
|x + y| ≤ |x| + |y|.
Proof.
|x + y|
2
= (x + y) · (x + y)
= |x|
2
+ 2x · y + |y|
2
≤ |x|
2
+ 2|x||y| + |y|
2
= (|x| + |y|)
2
.
So
|x + y| ≤ |x| + |y|.